Richness

In addition to Shannon diversity, reported in the manuscript, we conducted the analysis using richness as our focal diversity metric. During the with seed rain stage, external seed addition is added to the simulated plant communities. This external seed addition is calculated to be 100% of the average monoculture seed rain across all species. Therefore, even rare species are able to maintain low population sizes. For this reason, there is often no slope within each planted species richness treatment. In other words, all communities that began with 32-species will continuously contain 32-species, because of seed addition. We thus report only the across-richness treatment models, omitting those within-richness treatments because they are invalid models.

Mirroring Shannon diversity in the manuscript, our models for the across-treatment effect were encoded as: Biomass ~ -1 + Stage + Stage:Richness. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.

1 Figure 2 - Fishbone pattern

The relationship between richness and total biomass was qualitatively similar to that of Shannon diversity. The direction and magnitude of the relationship between richness a total biomass are consistent across all of the models. The only variation between results is that in Forest2, the significance of the seed rain and no seed rain estimates both change; the seed rain slope becomes significant, though maintaining almost no slope, and further the no seed rain phase becomes insignificant, though again maintaining its general slope.

2 Figure 3 - Across-treatment effect

Again, the general structure of the relationship between the communities’ underlying coexistence dynamics and their emergent BEF relationship are maintained. The only qualitative difference is that the slope of the interaction is significantly less steep during the seed rain phase. This results stems from richness not fully capturing the changes in species composition within the communities. Because seed addition ensures most species are likely present within the plots, richness during the seed rain phase is unlikely to change.

3 Model validation

This section of the document describes the statistical models’ validation, using richness as the focal biodiversity metric and total biomass as the focal ecosystem function.

Important terms:

3.1 Grass1

Clark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.

3.1.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Richness 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                23.04      1.37    20.34    25.70 1.00     2573     2369
## StageWithoutseedrain              6.06      1.67     2.73     9.28 1.00     2327     2505
## StageWithseedrain:Richness        1.92      0.10     1.73     2.12 1.00     2368     2241
## StageWithoutseedrain:Richness     7.03      0.31     6.45     7.62 1.00     2590     2342
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    18.61      0.48    17.72    19.56 1.00     3731     2812
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error     Q2.5     Q97.5
## R2 0.5464343 0.01616722 0.513239 0.5760942

3.1.2 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

3.1.2.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.1.2.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

3.1.2.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

3.2 Grass2

Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.

3.2.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Richness 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                60.23      0.96    58.33    62.13 1.00     2863     2971
## StageWithoutseedrain             60.29      0.96    58.43    62.16 1.00     2637     2930
## StageWithseedrain:Richness        0.66      0.06     0.53     0.78 1.00     2990     2743
## StageWithoutseedrain:Richness     0.87      0.06     0.75     1.00 1.00     2795     3117
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    13.85      0.36    13.20    14.56 1.00     3757     2652
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2838514 0.02308112 0.2382054 0.3282565

3.2.2 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

3.2.2.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.2.2.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

3.2.2.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

3.3 Grass3

May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.

3.3.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Richness 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                50.11      0.71    48.72    51.49 1.00     2202     2346
## StageWithoutseedrain             49.40      0.77    47.86    50.90 1.00     2538     2406
## StageWithseedrain:Richness        0.26      0.05     0.17     0.34 1.00     2521     2422
## StageWithoutseedrain:Richness     0.48      0.08     0.32     0.65 1.00     2584     2522
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.87      0.26     9.37    10.39 1.00     3770     2678
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5     Q97.5
## R2 0.08364221 0.01798936 0.05061246 0.1215145

3.3.2 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

3.3.2.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.3.2.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

3.3.2.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

3.4 Forest1

Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.

3.4.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Richness 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                42.31      1.52    39.32    45.28 1.00     2525     2306
## StageWithoutseedrain             38.57      1.82    35.01    42.13 1.00     2707     2646
## StageWithseedrain:Richness        1.21      0.14     0.94     1.48 1.00     2691     2643
## StageWithoutseedrain:Richness     3.72      0.53     2.70     4.75 1.00     2497     2409
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    21.32      0.53    20.34    22.39 1.00     3861     2536
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.1493185 0.02177845 0.1068745 0.1922317

3.4.2 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

3.4.2.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.4.2.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

3.4.2.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

3.5 Forest2

Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.

3.5.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Richness 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                30.87      0.73    29.44    32.31 1.00     2551     2759
## StageWithoutseedrain             23.49      0.91    21.69    25.24 1.00     2480     2764
## StageWithseedrain:Richness       -0.00      0.05    -0.10     0.09 1.00     2550     2724
## StageWithoutseedrain:Richness    -1.22      0.17    -1.56    -0.88 1.00     2355     2323
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    10.59      0.27    10.08    11.15 1.00     3515     2837
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2986958 0.02404684 0.2513005 0.3456881

3.5.2 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

3.5.2.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.5.2.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

3.5.2.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

3.6 Dryland

Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.

3.6.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Richness 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                76.08      0.64    74.83    77.37 1.00     2698     2493
## StageWithoutseedrain             80.36      0.91    78.56    82.11 1.00     2441     2587
## StageWithseedrain:Richness        0.04      0.04    -0.04     0.12 1.00     2851     2582
## StageWithoutseedrain:Richness    -1.81      0.30    -2.39    -1.23 1.00     2383     2458
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.20      0.23     8.75     9.65 1.00     3151     2850
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5      Q97.5
## R2 0.05185961 0.01469274 0.02567783 0.08352739

3.6.2 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

3.6.2.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.6.2.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

3.6.2.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.